There is no dedicated official Github client for Linux at this time.
There are, however, several GUI Git clients that will work with Github. I know you say you're not looking for Git clients, so if you're fully committed to that stance your answer is in the first sentence of this post. However, it's important to note that Github is Git, just a hosted version of it with some minor tweaks. You can use any Git client to interact with it, and there are many GUI clients that give a similar polish and ease of use to the process as you would expect from the official client.
There are many options, but of them all I recommend SmartGit for having a good user experience and good Github-specific integration (supporting Github-style pull requests and comments). It's the one I use for Linux when not using the command line. The diff/merge view is really nice.
You may also prefer GitEye if your focus is more on integrating GitHub's issue tracking. However, that's the extent of their GitHub-specific support.